The class blog for Math 3010, fall 2014, at the University of Utah

Tag Archives: Emmy Noether

After our discussion in class about the work of Sophie Germain, I was interested in learning more about other prominent women in mathematics. I’m sure we will go over some of them in class, but here is what I discovered about some very smart women.

One of the earliest known female mathematicians was Hypatia. She lived in the time period of approximately 350-416 C.E. She was excellent at mathematics, astronomy and philosophy. No doubt this is because her father was Theon, one of the last members of the library of Alexandria. Unfortunately for us, we do not know many of her contributions to science. She is more well known for her brutal death. She was riding in her carriage, when she was forcefully removed, stripped, beaten to death, and then her body was burned. Not a nice way to go. Regardless, of that cruelty, she is one of the first well known women mathematicians, and in her time that was quite an accomplishment.

Another leading lady in mathematics was Ada Lovelace. She lived from 1815-1852 as the daughter of well known writer, Lord Byron. She never met her father, and her mother advocated her to study fields that were different from language and poems. Essentially, anything different from what her father was well known for. It must have been a bad break up. Thus, math and science it was. Turns out, she is credited with being the world’s first programmer. But before that achievement, she demonstrated ingenuity as a child. She set her mind toward the daunting task of flying, at the young age of twelve. She researched materials, how to build wings, and even wanted to incorporated steam! Being curious from a young age really inspired her to continue her study of the sciences.

Because of the strict laws against the education of women she had to study mathematics with a tutor, she could not technically enroll in university. She met Charles Babbage later in life and their friendship encouraged her studies. They continued their correspondence even after her marriage to the Earl of Lovelace. At the time Babbage was working on a theoretical machine called the Analytical Engine. The idea was that the Engine could store numbers, and it could do long cycles and loops without the help of people. She wrote to Babbage about including Bernoulli numbers and how such implicit functions could be solved by the Engine. According to Wolfram Alpha, “The Bernoulli numbers are a sequence of signed rational numbers that can be defined by the exponential generating function. These numbers arise in the series expansions of trigonometric functions, and are extremely important in number theory and analysis.” In order to calculate Bernoulli numbers, there must be a lot of operations involved. To top it off, they anticipated that the Analytical Engine could perform this task. Below I have pictured one of Ada’s tables on how she envisioned the Engine could compute this. Remarkably enough, Lady Lovelace once said, “The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform. It can follow analysis; but it has no power of anticipating any analytical relations or truths. Its province is to assist us in making available what we are already acquainted with.” She understood that the machine is only as good as the people who are using it. It cannot come up with new ideas, or understand why it is doing computation, it can only do said computation. If this machine were to have been made, it would have been an incredible invention. However, the fact that it was never brought to production, does not in any way reduce all of the work both Ada and Charles did.

Unfortunately, there has been speculation that Ada did not contribute in the mathematical sense, but was merely a notetaker for Babbage. This is baffling because in his autobiography, Babbage gives her credit for all of the theoretical math she did for his Analytical Engine. I could continue this post with a commentary about women in science even today, but I’d better move onto the final female mathematician I wish to recognize.

The final female mathematician I wish to discuss is Emmy Noether. Emmy was born in Germany in the late 1800’s. She was denied a lot of formal education because she was a woman. She began her studies with piano and languages, but soon discovered a passion for math, like her father, and her brother. Universities in Germany were hesitant to let her become a professor, although, she did get the status of Associate Professor eventually. This title was taken away however, when the Nazi’s came to power because she was Jewish. Despite all of this, she had many notable accomplishments. So much so, that Albert Einstein once referred to her as “the most significant creative mathematical genius thus far produced since the higher education of women began.” This is high praise, especially coming from a man our society reveres as the most intelligent man ever known.

She was behind a revolutionary theorem, called Noether’s Theorem. This theorem states that: “Each symmetry of a system leads to a physically conserved quantity. Symmetry under translation corresponds to conservation of momentum, symmetry under rotation to conservation of angular momentum, symmetry in time to conservation of energy, etc.” And when I first read this, I was quite confused. However, with some help from my sources, I was able to wrap my mind around it to a certain extent. Noether is telling us that when we find symmetrical things, in nature or otherwise, there is some sort of conservation force that goes with it. One example of this, that is referenced in the New York Times article, is the relationship between time and energy. To paraphrase, if a person throws a ball up in the air right now, or throws it the same way sometime in the future, the time does not affect the trajectory of the ball. This means that the symmetry of time is related to the conservation of energy. This is crucial to how we think about physics today, and I could definitely relate this to my old physics teacher being like a broken record and telling us energy cannot be created or destroyed, it only changes form. Emmy clearly made an impact on not only math, but the way we think about certain concepts today. She even developed some of the mathematical formulas that Einstein used for his Theory of Relativity.

It seems to me that Emmy deserves much more recognition than she is receiving. Truthfully, I had not even heard of her until I began research for this blog post. I know this is not a class about how our society can improve, but one way would be to get more women in math and science. It is interesting to think about how limited women once were. I am optimistic about the progress we have made in that regard, but just think about how much further along we could possibly be in terms of figuring out the mysteries of the world if we had help from every person, from every demographic, and every gender. I do not know if this is possible, but inclusion is a nice thought. These ladies kicked butt in their time, and I hope that the women of the present and the future follow their example and continue to do the same.

I have recently learned that October 14th was Ada Lovelace Day! Ada Lovelace Day celebrates women in all areas of science. And because of that, I would like to dedicate this post to all the amazing ladies out there making leaps and bounds in the sciences. You are an inspiration to me, but all young women of the world.

As I have gone through the process of gaining a higher education in mathematics, I have made a startling realization that I am alone. Sure, there are other women in my math classes, but the majority of the students are men. I have had to rely on my own strength and diligence to get through the challenging courses. When I started working on my degree, I had many counselors and professors that discouraged me from entering into such a field due to the fact that it was challenging, and the odds were I would not succeed. Whether this opinion was developed from me being a female or not, I have a hard time believing that a male would receive that same type of consolation. Also, it is a popular belief in our culture here in Utah that most women should not enter into the fields of science and mathematics, and are better off obtaining degrees that will benefit them as homemakers. Hence, most women do not pursue a degree in mathematics or science. It troubled me to know that there are no women that I could turn to for help in my field of choice. In my History of Mathematics class, we have been learning about the great minds of mathematics which have mostly been men. However, last week in class, we learned about Sophie Germain, a woman mathematician. This got me thinking that I have never before heard, or learned about other women in the field of mathematics. I’ve been asking myself, why don’t I know more about these women? So I decided to do some research and find other women who have contributed to the field of mathematics and made it possible for other women, like myself, to gain a higher education.

One of the first known female mathematicians was Hypatia (370-415 A.D.). Her father was a well-educated man, and Hypatia spent a lot of time in the world of education learning from her father. From her father’s teachings, Hypatia become very educated in math, science, and astronomy and would impart this knowledge to students in her home. Large crowds would also come and listen to her teach in the streets. Her fame and popularity, however, turned to be her downfall as she was killed by Christian zealots.

Sophie Germain (1776-1831) was born in a time of revolution, which was shown in her character. During this time, it wasn’t socially acceptable for women to have access to the same education as men. This didn’t stop Sophie from becoming a great mathematician, and being the first woman to win a prize from the French Academy of Sciences for her work on the theory of elasticity. It should be noted that during her life she often worked under a false name to avoid persecution for breaking social boundaries of women in education.

Sofia Kovalevskaya (1850-1891) was born in Russia, where women were not allowed to attend universities. In order for her to pursue some type of higher education, she decided to get married so she could travel to Germany, and was able to be privately tutored by a professor. Sofia was granted a PhD, and went on to produce wonderful works in the fields of mathematics and science, but was always faced with adversity. Despite her hardships, her contributions were vast, and she expanded the opportunities for women in education and women’s rights.

Emmy Noether (1882-1935) grew up in Germany, where she wasn’t allowed to receive a university education. Growing up, she was educated in language, and the common tasks expected from women. At age eighteen, she decided to take courses in mathematics, and was able to become a university student. She received a PhD, and became an unofficial associate professor at the University of Göttingen. However, in 1933 she lost that title because she was Jewish. She decided to move to America and became a lecturer and researcher. There she developed many of the mathematical foundations for Einstein’s general theory of relativity. Einstein later wrote of her that she was “the most significant creative mathematical genius thus far produced since the higher education of women began”(Zielinski).

Ingrid Daubechies (1954-Present) is the first female president of the International Mathematical Union, and is a strong advocate for women in science and mathematics. As a girl, she studied physics and eventually received her PhD, along with other awards. Her most important discovery was in the field of wavelets, which are “mathematical functions useful in digital signal processing and image compression as well as in many other branches of applied and pure mathematics”(Riddle). In a recent interview Daubechies was asked why there is the assumption that men are better at mathematics than women. Her response to this question, “I disagree with this view – completely. There is a highly variable percentage of women in academia and in departments of mathematics across Europe. Differences are so enormous that it becomes obvious that it has something to do with cultural habits, which differ from one nation to another, and not with intelligence”(TWAS).

In conclusion, there have been many women who have made significant contributions to the fields of math and science, and have influenced the works of other male scholars. As a woman in higher education and mathematics, I admire the hardships and work these women accomplished, and wish that more was said about them. In doing this research, I’ve realized I am not alone, and I have many great examples of women who have worked hard and overcame societal obstacles. As a future teacher, I aspire to influence more girls to pursue college degrees and not be intimidated by the “male dominated” subjects, and realize that women are just as intelligent as men.